Showing papers in "Rocky Mountain Journal of Mathematics in 1976"

Positive definite functions and generalizations, an historical survey

Abstract: 2 fi COS Xi + i X 6 S i n i = °i = l ' ' i = l ' Likewise it is easily verified directly that e is p.d. for real \\ , but it is not so straightforward to see that such functions as e~H e~*, and (1 4x)\" ? e p.d. These and other examples are discussed in § 3. Positive definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics since the beginning of this century. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, embedding problems, information theory, and other areas. Their history constitutes a good illustration of the words of Hobson [51, p. 290] : \"Not only are special results, obtained independently of one another, frequently seen to be really included in

Some probabilistic problems and methods in singular perturbations

Abstract: We discuss in detail the asymptotic analysis of deterministic and stochastic problems, within a certain class, from the point of view of order reduction or contraction of description.

Categories of actions and Morita equivalence

Abstract: 0. Introduction. Let A be a small category. By a (right) A-action is meant a (covariant) functor A—• Ens, where Ens is the category of sets, and by a morphism of A-actions a morphism of the corresponding functors. The category of A-actions will be denoted by Ens, and A will be called the operator category. An abstract characterization of categories of actions is given in [3]. Two (small) categories A and B are called Morita-equivalent (A—MB) if their categories of actions are equivalent in the usual sense (Ens — Ens); i.e., if there exist functors F and G between Ens and Ens such that FG and GF are isomorphic to the corresponding identity functors. The following questions arise: I. How can we characterize Morita equivalence intrinsically? II. How can we construct all categories Morita-equivalent to a given one? Answers are given to I and II in [5], when the operator categories are (finite) groups (in which case Morita equivalence implies isomorphism), and in [1] and [7], when they are arbitrary monoids. The situation is analogous to the Morita theory for modules over a ring, described, e.g., in [2], [4]. In the present paper we provide answers to I and II for the case of arbitrary operator categories. One form of the answer can be stated in terms of the notion of weak equivalence —w between categories. A weak functor f : A—* B is like an ordinary functor, but without the requirement that the image of an identity morphism be an identity morphism (it is then perforce an idempotent). There is a corresponding notion of weak fiinctor morphism, and the result can be stated (Theorem 4.4):

The brauer group of a ring modulo an ideal

Abstract: Let R be a commutative ring. Let B(R) be the Brauer group of R as defined in [2]. If/ is an ideal of R the natural homomorphism from fl to RII induces a homomorphism from B(R) to B(RII). We study this homomorphism in two contexts. In § 1 we show that if / is a nilpotent ideal then the homomorphism from B(R) to B(RjI) is an isomorphism. In § 2 we assume R is a local ring and I is its maximal ideal. We describe the kernel of the homomorphism from B(R) to B(RII) and then show this homomorphism is onto if RJI is a number field, the p-adic completion of a number field, or a function field in one variable over a finite field. The results of § 1 have been proved by R. Hoobler using cohomological methods, but our proof is algebra theoretic and uses recent results of E. Ingraham [8]. Professor Ingraham provided part °f the argument used in the proof of Theorem 1. The results of $ 2 include some of those of V. J. Katz [9] who assumed R to be a regular local domain. Throughout, our notational conventions will be the same as in [7] which contains the definitions of all unexplained terminology.

On the existence of solutions to multipoint boundary value problems

Abstract: We are concerned with the nth order differential equation t/ = f(x, y, t/, •••, t/ (n -1 ) where it is assumed throughout that / is continuous on [ a, ß) X fl, « < ß ^ oo f and that solutions of initial value problems are unique and exist on [a,ß). Our main concern will be to show that under certain conditions the uniqueness of solutions of multipoint boundary value problems implies the existence of solutions of such problems.

Lattice-valued Borel measures

Abstract: Let X be a completely regular T1 space, E a boundedly complete vector lattice, C(X) (Cb(X)) the space of all (all, bounded), real-valued continuous functions on X. In order convergence, we consider E-valued, order-bounded, -additive, -additive, and tight measures on X and prove some order-theoretic and topological properties of these measures. Also for an order-bounded, E-valued (for some special E) linear map on C(X), a measure representation result is proved. In case E n separates the points of E, an Alexanderov's type theorem is proved for a sequence of -additive measures.

Oscillation and even order linear diΘerential equations

Abstract: Let 9 be a continuous function from [0, oo ) to (0, oo ), and let n be a positive integer. We shall obtain herein oscillation and nonoscillation criteria for (1) u<"> + qu = 0 on [0, oo ). It follows from results of J. G. Mikusinski [7] and I. T. Kiguradze [5] (see also G. V. Anan'eva and V. I. Balaganskii [1], H. C. Howard [4], V. A. Kondrat'ev [6], and C. A. Swanson [8, Theorem 4.59, p. 173] ) that if 0 < a < 2n 1 and Jo Fq(t) dt = oo? then every solution of (1) is oscillatory (i.e., every solution of (1) has an unbounded set of zeros). Thus, we shall assume throughout that if 0 < a < 2n — 1, then Jo tq(t) dt < oo . Also, since our primary results compare (1) to certain second order equations, we assume n = 2. R. Grimmer [2] has recently done some related work comparing higher order equations to second order equations. THEOREM 1. Suppose the second order equation

Mellin convolutions and H -function transformations

Abstract: ABSTRACT. An elegant expression is obtained for an Hfunction transform of the Mellin convolution of two functions in terms of the Mellin convolution of H-function transforms of the functions. This main result leads to several new and interesting relations involving, for instance, Fourier, Hankel, Laplace, Stieltjes, Whittaker, and K-transforms. Some of the special cases discussed provide useful additions to known tables of these integral transforms.

On the oscillation of a class of fourth order differential equations

Abstract: 1. Introduction. This paper is concerned with fourth order dif­ferential equations of the form (L) (p(x)y")" - q(x)y" - r(x)y = 0, where p, q and r are assumed to be continuous, real-valued functions on the interval [a, °° ). In addition, it will be assumed throughout that p > 0, q ^ 0 and r^Oon [a, o° ), with r not identically zero on any subinterval. If q is a (non-negative) constant, then (L) is self-adjoint; otherwise (L) is non-self-adjoint. The objective of the paper is to study the oscillatory behavior of the solutions of (L). A non-trivial solution y is oscillatory if the set of zeros of y is not bounded above. If the set of zeros of y is bounded above, which implies y has only finitely many zeros, then y is non-oscillatory. Hereafter, the term "solution" will be interpreted to mean non-trivial solution. Various special cases of (L) have been studied in detail. In par­ticular, we refer to the fundamental work of W. Leigh ton and Z. Nehari [5, Part I] on the self-adjoint equation (1) (P(x)y")" - r(x)y = 0. M. Keener [3, Part I] continued the investigation of (1), concentrat­ing on the oscillatory behavior of solutions. S. Hastings and A. Lazer [2] considered the self-adjoint equation (2) y(4) - r(x)y = 0, showing that (2) has a linearly independent pair of bounded oscillatory solutions when it is assumed that r E C ' [a, «> ), with r > 0 and r ' ^ 0 on [a, oo ). S. Ahmad [1] has also studied (2), giving necessary and sufficient conditions for the existence of a linearly independent pair of oscillatory solutions. Finally, we refer to the work of V. Pudei [6], [7] in which the equation (3) yW - q(x)y" - r(x)y = 0 is considered.